PRIMARY OBJECTIVES: I. To assess the safety of cellular immunotherapy utilizing ex vivo expanded autologous central memory T cell (TCM)-enriched cluster of differentiation (CD)8+ T cells genetically-modified to express a CD19-specific chimeric antigen receptor (CAR) in conjunction with a standard myeloablative autologous hematopoietic stem cell transplantation (aHSCT) for research participants with high-risk intermediate grade B-lineage non-Hodgkin lymphomas who have relapsed after primary therapy, or who did not achieve complete remission with primary therapy. (Phase I)